Manifolds with small Heegaard Floer ranks

Hedden, Matthew; Ni, Yi

doi:10.2140/gt.2010.14.1479

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Manifolds with small Heegaard Floer ranks

Matthew Hedden and Yi Ni

Geometry & Topology 14 (2010) 1479–1501

DOI: 10.2140/gt.2010.14.1479

Abstract

We show that the only irreducible three-manifold with positive first Betti number and Heegaard Floer homology of rank two is homeomorphic to zero-framed surgery on the trefoil. We classify links whose branched double cover gives rise to this manifold. Together with a spectral sequence from Khovanov homology to the Floer homology of the branched double cover, our results show that Khovanov homology detects the unknot if and only if it detects the two component unlink.

Keywords

Heegaard Floer homology, Khovanov homology, two-component unlink, torus bundle

Mathematical Subject Classification 2010

Primary: 57M27

Secondary: 57M25

References

Publication

Received: 11 August 2009

Revised: 26 May 2010

Accepted: 24 February 2010

Published: 11 June 2010

Proposed: Ron Fintushel

Seconded: Peter Ozsváth, Ron Stern

Authors

Matthew Hedden
Department of Mathematics Michigan State University East Lansing, MI 48824
http://www.math.msu.edu/~mhedden/Site/Home.html
Yi Ni
Department of Mathematics Caltech Pasadena, CA 91125
http://www.its.caltech.edu/~yini/

Volume 14, issue 3 (2010)